Single-Variable Calculus I

Purpose of Course  showclose

This course is designed to introduce you to the study of Calculus.  You will learn concrete applications of how calculus is used and, more importantly, why it works.  Calculus is not a new discipline; it has been around since the days of Archimedes.  However, Isaac Newton and Gottfried Leibniz, two 17th-century European mathematicians concurrently working on the same intellectual discovery hundreds of miles apart, were responsible for developing the field as we know it today.  This brings us to our first question, what is today’s Calculus?  In its simplest terms, calculus is the study of functions, rates of change, and continuity.  While you may have cultivated a basic understanding of functions in previous math courses, in this course you will come to a more advanced understanding of their complexity, learning to take a closer look at their behaviors and nuances.

In this course, we will address three major topics: limits, derivatives, and integrals, as well as study their respective foundations and applications.  By the end of this course, you will have a solid understanding of the behavior of functions and graphs.  Whether you are entirely new to Calculus or just looking for a refresher on a particular topic, this course has something to offer, balancing computational proficiency with conceptual depth.

Course Information  showclose

Welcome to MA101.  Below, please find general information on this course and its requirements. 

Course Designer: Clare Wickman

Primary Resources: This course is comprised of a range of different free, online materials.  However, the course makes primary use of the following: Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials.  Pay special attention to Units 1 and 2, as these lay the groundwork for understanding the more advanced, exploratory material presented in the latter units; note that Unit 1 will mostly be a review.  You will also need to complete:
  • Unit 1 Assignments
  • Unit 2 Assignments
  • Unit 3 Assignments
  • Unit 4 Assignments
  • Unit 5 Assignments
  • Unit 6 Assignments
  • Unit 7 Assignments
  • Final Exam
Note that you will only receive an official grade on your final exam.  However, in order to adequately prepare for this exam, you will need to work through the quizzes and problem sets listed above.
 
In order to “pass” this course and earn your Saylor Foundation Course Certificate, you will need to earn a 70% or higher on the Final Exam.  Your score on the exam will be tabulated as soon as you complete it.  If you do not pass the exam, you may take it again.

Time Commitment: This course should take you approximately 93.75 hours to complete.  At the beginning of each unit, there is a detailed list of time advisories for each subunit.  These estimates factor in the time required to watch each lecture, work through each reading thoughtfully, and complete each assignment.  However, these should be seen as guidelines, not goals; each learner is different, and you may find that your pace changes throughout the course.  Mastery of the material, rather than strict adherence to the time estimates, is the measure of success in this course.  It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set a schedule for yourself.  For example, Unit 1 should take you 13 hours.  Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 2 hours) on Monday night; subunit 1.2 (a total of 3 hours) on Tuesday night; etc.

Tips/Suggestions: If a video lecture stops making sense to you, pause it—this is a luxury you only have in a course of this nature!—and return to the readings to get up-to-speed on the material.  Remember to note down the time at which you paused the video lecture, in case your browser times out.  Try to take notes on the resources, writing down any formulas or other information you need to know.  These notes will be useful as you study for your Final Exam. 

Learning Outcomes  showclose

Upon successful completion of this course, the student will be able to:
  • Define and identify functions.
  • Define and identify the domain, range, and graph of a function.
  • Define and identify one-to-one, onto, and linear functions.
  • Analyze and graph transformations of functions, such as shifts, dilations, and compositions of functions.
  • Characterize, compute, and graph inverse functions.
  • Graph and describe exponential and logarithmic functions.
  • Define and calculate limits and one-sided limits.
  • Identify vertical asymptotes.
  • Define continuity and determine whether a function is continuous.
  • State and apply the Intermediate Value Theorem.
  • State the Squeeze Theorem, and use it to calculate limits.
  • Calculate limits at infinity and identify horizontal asymptotes.
  • Calculate limits of rational and radical functions.
  • State the epsilon-delta definition of a limit, and use it in simple situations to show a limit exists.
  • Draw a diagram to explain the tangent-line problem.
  • State several different versions of the limit definition of the derivative, and use multiple notations for the derivative.
  • Describe the derivative as a rate of change, and give some examples of its application, such as velocity.
  • Calculate simple derivatives using the limit definition.
  • Use the power, product, quotient, and chain rules to calculate derivatives.
  • Use implicit differentiation to find derivatives.
  • Find derivatives of inverse functions.
  • Find derivatives of trigonometric, exponential, logarithmic, and inverse trigonometric functions.
  • Solve problems involving rectilinear motion using derivatives.
  • Solve problems involving related rates.
  • Define local and absolute extrema.
  • Use critical points to find local extrema.
  • Use the first and second derivative tests to find intervals of increase and decrease and to find information about concavity and inflection points.
  • Sketch functions using information from the first and second derivative tests.
  • Use the first and second derivative tests to solve optimization (maximum/minimum value) problems.
  • State and apply Rolle’s Theorem and the Mean Value Theorem.
  • Explain the meaning of linear approximations and differentials with a sketch.
  • Use linear approximation to solve problems in applications.
  • State and apply L’Hopital’s Rule for indeterminate forms.
  • Explain Newton’s method using an illustration.
  • Execute several steps of Newton’s method and use it to approximate solutions to a root-finding problem.
  • Define antiderivatives and the indefinite integral.
  • State the properties of the indefinite integral.
  • Relate the definite integral to the initial value problem and the area problem.
  • Set up and calculate a Riemann sum.
  • Estimate the area under a curve numerically using the Midpoint Rule.
  • State the Fundamental Theorem of Calculus and use it to calculate definite integrals.
  • State and apply basic properties of the definite integral.
  • Use substitution to compute definite integrals.

Course Requirements  showclose

In order to take this course you must:

√     Have access to a computer.

√     Have continuous broadband Internet access.

√     Have the ability/permission to install plug-ins or software (e.g., Adobe Reader or Flash).

√     Have the ability to download and save files and documents to a computer.

√     Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√     Be competent in the English language.

√     Have read the Saylor Student Handbook.

Unit Outline show close


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